Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Let (G = \langle g \rangle) be a cyclic group, and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (G) can be written as (g^n) for some integer (n). Because (H) is non-trivial, there exists a smallest ___ _______ _____ ______ _______ __________.
____ ___ _________ ______ _______ ________ _____ ____ ________ ______.
______ _____ ________ _______ ______ ___.
___ _____ ____ _________ ___ _________ _________ ________ _________.
_______ _______ _________ ________ __________ _________.
___ __________ __________ _____ _______ ___ __________ ______ _________ ____.
_________ _________ ____ _________ ______ __________ _________ _________.
________ __________ __________ ____ ______ _____ __________ ______ _____ __________.
______ ______ _______ ___ _________ ________ ________ _____.
______ ___ _____ ____ ________ ___.
______ ___ _____ _______ ________ _________ __________ __________ ____ __________ ________ ____.
______ ____ _________ _______ __________ _____ ___ _______ _____ ____.
_______ ______ _________ _______ ____ __________ ___ _________ ___ _____.
________ ______ ____ __________ _________ _____ __________ __________ ______ ___ ________ ______.
____ __________ _____ ___ ___ ___ _________ ______ ___ _____.
___ ___ _____ ______ ______ __________ ______ __________ ___ _____.
_______ ___ ____ __________ ____ __________ __________ __________ ______ ______ ________ ____.
_________ _________ ____ _____ _________ _______ ________ ___ ____ _______ _______.
________ ________ ___ ____ ____ _____ _____.
__________ _________ ____ ______ ______ ______.
Get Full Answer on WhatsApp
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
. Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
Find all the units of .
Check whether or not
is a field.
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
समूह समाकारिता के मूल प्रमेय की सहायता से निम्नलिखित प्रमेय, जिसे ज़ासनहाउस (तितली) प्रमेयिका कहा जाता है, को सिद्ध कीजिए।
यहाँ चित्र में दिए गए टेक्स्ट का लिखित रूप है:
मान लीजिए H और K समूह G के उपसमूह हैं और H' और K', H और K के प्रसामान्य उपसमूह हैं। तब
i) ,
ii) ,
iii)
क) क्रमविनिमेय वलय R की गुणजावली I के लिए,, किसी
के लिए परिभाषित कीजिए। दिखाइए कि
यहाँ चित्र में दिए गए टेक्स्ट का लिखित रूप है:
i) की एक गुणजावली है।
ii) .
iii) कुछ स्थितियों में
।
ख) क्या वलय R की कोई भी दो गुणजावलियों I और J के लिए,
? अपने
उत्तर की पुष्टि कीजिए।
(5)